Consider a pure exchange economy with two goods, z and y, and three consumers, 1, 2, and 3. The consumer's utility functions are respectively. The endowments are (3,1) for consumer 1, (3,6) for consumer 2, and (2, 4) for consumer 3. (a) Find the aggregate excess demand for each good. You may use Marshallian demands derived earlier in the course. demand for Good 1 is Solution: If the prices of the two goods are p₁ and p2, then the three consumers have wealths 3p1 +p2, 3p1 +6p2, and 2p1 +4p2 respectively. Using the Marshallian demands for Cobb-Douglas utilities with these levels of wealth, we find that the aggregate excess and for Good 2 is u¹(x, y) = x²y u²(x, y) = xy² u²(x,y)= 21(P₁.P₂) 22(p1, p2) =xy = 6p1+2p2 3pi+6p2 2p1 +4p2 + 3p₁ 3p₁ 2p1 14p2 3p1 + 3p₁+P2 6p₁+12p₂, 2p₁ +4p₂ 3p₂ 2p₂ 3p₂ 4p₁ 14 P₂ (b) Show that your answer for part (a) satisfies Walras' Law. Solution: For each (p1, p2), we have -8 11 P₁²1 (P₁-P₂) + P2²2(P₁-P2) = 14p²-4p₁ +4P₁ P₂ = 0. (e) Find all Walrasian equilibria for the given endowments. ·$P₁. Solution: From the market-clearing condition 21(P₁-P2) = 0, we obtain p2 = Substituting into the demands gives the allocation ¹ = (#), 2²- 23=(). This is the only equilibrium except for multiples of the prices by a common (4.), and factor
Consider a pure exchange economy with two goods, z and y, and three consumers, 1, 2, and 3. The consumer's utility functions are respectively. The endowments are (3,1) for consumer 1, (3,6) for consumer 2, and (2, 4) for consumer 3. (a) Find the aggregate excess demand for each good. You may use Marshallian demands derived earlier in the course. demand for Good 1 is Solution: If the prices of the two goods are p₁ and p2, then the three consumers have wealths 3p1 +p2, 3p1 +6p2, and 2p1 +4p2 respectively. Using the Marshallian demands for Cobb-Douglas utilities with these levels of wealth, we find that the aggregate excess and for Good 2 is u¹(x, y) = x²y u²(x, y) = xy² u²(x,y)= 21(P₁.P₂) 22(p1, p2) =xy = 6p1+2p2 3pi+6p2 2p1 +4p2 + 3p₁ 3p₁ 2p1 14p2 3p1 + 3p₁+P2 6p₁+12p₂, 2p₁ +4p₂ 3p₂ 2p₂ 3p₂ 4p₁ 14 P₂ (b) Show that your answer for part (a) satisfies Walras' Law. Solution: For each (p1, p2), we have -8 11 P₁²1 (P₁-P₂) + P2²2(P₁-P2) = 14p²-4p₁ +4P₁ P₂ = 0. (e) Find all Walrasian equilibria for the given endowments. ·$P₁. Solution: From the market-clearing condition 21(P₁-P2) = 0, we obtain p2 = Substituting into the demands gives the allocation ¹ = (#), 2²- 23=(). This is the only equilibrium except for multiples of the prices by a common (4.), and factor
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Consider a pure exchange economy with two goods, z and y, and three consumers, 1, 2, and
3. The consumer's utility functions are
respectively. The endowments are (3,1) for consumer 1, (3,6) for consumer 2, and (2, 4) for
consumer 3.
(a) Find the aggregate excess demand for each good. You may use Marshallian demands
derived earlier in the course.
Solution: If the prices of the two goods are p₁ and p2, then the three consumers have
wealths 3p1 +p2, 3p1 +6p2, and 2p1 +4p2 respectively. Using the Marshallian demands
for Cobb-Douglas utilities with these levels of wealth, we find that the aggregate excess
demand for Good 1 is
21(P₁.P2)-
=
and for Good 2 is
u¹(x, y) = x²y
u²(x, y) = xy²
u³(z. y) = xy
=
22(P1, p2) =
=
6p1+2p2 3p1+6p22p1+4p2
3p1
14p2
3p₁
3p₁ + P2
3p₂
+
4P₁ 14
P2 3
3p₁
6p₁ + 12p2 2p₁ +4P2
+
3p₂
2p2
(b) Show that your answer for part (a) satisfies Walras' Law.
Solution: For each (p1, p2), we have
2p₁
8
=
-11
P1²1(P₁-P2) + P2²2(P₁-P2) = 14P² − 4p₁ +4P₁ - P2 =
14
= 0.
(e) Find all Walrasian equilibria for the given endowments.
Solution: From the market-clearing condition 21(P1, P2) = 0, we obtain p₂ =
Substituting into the demands gives the allocation 2¹ (₁), z² = (), and
23=(). This is the only equilibrium except for multiples of the prices by a common
factor.
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